The first five period-doubling points of the logistic map are , ,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Nonlinear dynamics and chaos

Computational Physics, Ch When trajectories separate: the driven pendulum and the logistic map

Problem

The first five period-doubling points of the logistic map xn+1=rxn(1−xn)x_{n+1}=rx_n(1-x_n) are r1=3.000000r_1=3.000000, r2=3.449490r_2=3.449490, r3=3.544090r_3=3.544090, r4=3.564407r_4=3.564407 and r5=3.568759r_5=3.568759, each quoted to six figures. ((a)) Form the four gaps gn=rn+1−rng_n=r_{n+1}-r_n and then the three ratios g1/g2g_1/g_2, g2/g3g_2/g_3 and g3/g4g_3/g_4, each to five figures. ((b)) Take the mean of the three ratios and compare it with the accepted Feigenbaum ratio δF=4.669202\delta_{\rm F}=4.669202. By how much do the two differ, and how does that difference compare with the scatter of the three ratios about their own mean? ((c)) A value quoted to six figures carries an uncertainty of about 5×10−75\times10^{-7}. Propagate that through g3/g4g_3/g_4, the ratio built from the two smallest gaps, and say what limits this measurement of δF\delta_{\rm F}: the precision of the bifurcation points, or the fact that the sequence of ratios was stopped at n=4n=4.

Browse the problem library